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The First Law for Slowly Evolving Horizons PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 1 9JANUARY2004 The First Law for Slowly Evolving Horizons Ivan Booth* Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland and Labrador, Canada A1C 5S7 Stephen Fairhurst† Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 (Received 5 August 2003; published 6 January 2004) We study the mechanics of Hayward’s trapping horizons, taking isolated horizons as equilibrium states. Zeroth and second laws of dynamic horizon mechanics come from the isolated and trapping horizon formalisms, respectively. We derive a dynamical first law by introducing a new perturbative formulation for dynamic horizons in which ‘‘slowly evolving’’ trapping horizons may be viewed as perturbatively nonisolated. DOI: 10.1103/PhysRevLett.92.011102 PACS numbers: 04.70.Bw, 04.25.Dm, 04.25.Nx The laws of black hole mechanics are one of the most Hv, with future-directed null normals ‘ and n. The ex- remarkable results to emerge from classical general rela- pansion ‘ of the null normal ‘ vanishes. Further, both tivity. Recently, they have been generalized to locally the expansion n of n and L n ‘ are negative. defined isolated horizons [1]. However, since no matter This definition captures the local conditions by which or radiation can cross an isolated horizon, the first and we would hope to distinguish a black hole. Specifically, second laws cannot be treated in full generality. Instead, the expansion of the outgoing light rays is zero on the the first law arises as a relation on the phase space of horizon, positive outside, and negative inside. Addition- isolated horizons rather than as a truly dynamical rela- ally, the ingoing null rays are converging. As we will see, tionship. Even existing physical process versions of the these horizons can be spacelike or null and include both first law [2,3] consider transitions between infinitesimally equilibrium and nonequilibrium states—an important separated isolated (or Killing) horizons. In this Letter, we feature for our perturbative study. By comparison, if introduce a framework which allows us to extend the first one is interested only in the dynamical phase, the space- law to all slowly evolving horizons, even those whose like dynamical horizons recently introduced by Ashtekar initial and final states are not infinitesimally separated. and Krishnan [5] are more relevant. However, such hori- Additionally, we provide a simple characterization of zons are always expanding and thus not so suitable for how close a horizon is to equilibrium, which will be studying transitions to and from equilibrium. useful in the final stages of numerical simulations of Hayward [4] has extensively studied the properties of black hole collisions. To obtain such a law, we first need FOTHs. Here, we summarize only those of his results a local dynamical definition of a black hole horizon. which are important for our work. First, consider the Hayward has introduced future outer trapping horizons quantities associated with the null vector fields ‘ and n, (FOTHs) for exactly this purpose. Furthermore, he has which we normalize so that ‘ n ÿ1. We denote their shown that they necessarily satisfy the second law—their relative expansions ‘ and n. Their twists are zero area cannot decrease in time [4]. since they are normal to the Hv cross sections of the ‘ n In this Letter, we will show that both the zeroth and first horizon. Finally, we write their shears as ab and ab . laws are also applicable to FOTHs. In order to do so, we Next, for each choice of the fields ‘ and n, there exists a introduce dynamical notions of surface gravity and an- scalar field C on H so that gular momentum which are applicable to all such hori- a ‘a ÿ Cna and ‘ Cn (1) zons. It follows immediately that the surface gravity is V a a a necessarily constant if the horizon is in equilibrium (i.e., are, respectively, tangent and normal to the horizon. Note isolated). Next, we introduce the notion of a slowly evolv- that V V ÿ 2C. Hayward [4] has shown that, ing horizon, for which the gravitational and matter fields if the null energy condition holds, then are slowly changing. It is only in this limited context that we expect to obtain the dynamical first law.We will show C 0 (2) that this is indeed the case. on a FOTH. Thus, the horizon must be either spacelike or Let us begin by recalling Hayward’s definition. null, and the second law of trapping horizon mechanics Future outer trapping horizon.—A future outer trap- follows quitep easily. If q~ab is the two metric on the cross ping horizon (FOTH) is a smooth three-dimensional sections, and q~ is the corresponding area element, then submanifold H of space-time which is foliated by a p p preferred family of spacelike two-sphere cross sections L V q~ ÿC n q~: (3) 011102-1 0031-9007=04=92(1)=011102(4)$20.00 2004 The American Physical Society 011102-1 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 1 9JANUARY2004 By definition, n is negative and we have just seen that C which can be written as is non-negative. Hence, we obtain the local second law:p If b the null energy condition holds, then the area element q~ !a :ÿnbr ‘ ; (4) of a FOTH is nondecreasing along the horizon [4]. a Clearly, on integration, the same law applies to the total where the arrow signifies that the derivative is pulled back area of the horizon. It is nondecreasing and remains to the horizon. Then, the isolated horizon surface gravity constant if and only if the horizon is null. is given by ‘a! while angular momentum infor- To further restrict the rescaling freedom of the null ‘ a mation is contained in the other components of ! . vectors, we require that v 1, where v is the folia- a L V Now, ! written in this form [Eq. (4)] continues to be tion parameter labeling the cross sections. Thus, choosing a an intrinsically well-defined quantity for trapping hori- the foliation parameter fixes the length of a, a, and the V zons. Then, an obvious generalization is to define null vectors. However, at this stage, we are still free to a change the foliation labeling and, by doing so, rescale V a a b v : V !a ÿnbV raV and and the null vectors by functions of v. From the isolated Z 1 p (5) horizon perspective, the normalization we have chosen is : 2 a a a J’ d x q~’ !~a; very natural, as in that case V ‘ , and it is customary 8 Hv to choose the foliation parameter v, so that n ÿdv. Physical characterization of trapping horizons.—The where ’a is any vector field tangent to the horizon cross laws of black hole mechanics are given in terms of sections, and !~a is the projection of !a into the two- a physical quantities such as energy, angular momentum, surface Hv. In the case where ’ is divergence free, J’ is surface gravity, and area. One of the advances of the independent of the normalization of ‘ and equal to other isolated horizon formalism was to provide definitions of popular expressions for angular momentum (such as the all these quantities at the horizon, without reference to Komar, Brown-York [6], or Ashtekar-Krishnan [5] defi- spacelike infinity or the space-time in which the horizon nitions). It is clear that both the surface gravity v and is embedded [1,2]. In generalizing these definitions to angular momentum J’ expressions reduce to their iso- FOTHs, we will be motivated by two requirements: (i) the lated horizon values if the horizon is null. new definitions should match the old ones when trapping The constraint law.—Consider the set of vector fields in horizons are isolated and (ii) the new definitions should H that generate one-parameter families of diffeomor- depend only on quantities that are intrinsic to the horizon, phisms that map two-dimensional cross sections Hv of and as such truly be properties of the horizon itself. That the horizon into each other. Any such vector field may be a a a said, the ultimate justification for these expressions will written in the form X x0V x~ , for some function a be found in their utility in the calculations that follow. x0 v and vector field x~ that is everywhere tangent to an For isolated horizons, both surface gravity and angular appropriate cross section. Then, by integrating the aXb momentum were defined with respect to a one-form !a component of the Einstein equations over Hv, we obtain the following relationship: Z p p Z p Z p 1 2 a 2 a b 1 2 ‘ab nab d xvLX q~ x~ LV q~!~a d x q~TabX d x q~ C LXq~ab 8G Hv Hv 16G Hv Z p p 1 2 d x2C q~ LX nC n LX q~: (6) 16G Hv A full derivation of this result will be given in [7], and in another paper we will see that this relation plays a crucial dE TdS work terms: (7) role in the Hamiltonian formulation of general relativity on manifolds with FOTHs as boundaries [8]. Here, we Furthermore, in the general case, there is no clear inter- show that in certain restricted situations Eq. (6) becomes pretation of the right-hand side of (6) as a flux of energy a first law for dynamical black holes. through the horizon. Instead, we will require the horizon Quasistationary horizons.—At first glance, one might to be ‘‘quasistationary’’ and then obtain a first law.
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